Abstract
In this study, we present a fractal generalized fourth-order Boussinesq equation which can describe the shallow water waves with the non-smooth boundary (such as the fractal boundary). Aided by the semi-inverse method, we establish its variational principle, which is proved to have a strong minimum condition via the He–Weierstrass theorem. Then, two powerful approaches namely the variational method (VM) and energy balance theory (EBT) are utilized to search for the periodic wave solutions. As expected, the results obtained by the two methods are almost the same. Furthermore, the impact of the fractal orders on the periodic wave structure is illustrated via the 3D plot and 2D curve. The results of this paper are expected to provide a reference for the study of periodic wave theory in fractal space.
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